Opuscula Math. 35, no. 3 (2015), 293-332
http://dx.doi.org/10.7494/OpMath.2015.35.3.293

Opuscula Mathematica

# Frames and factorization of graph Laplacians

Palle Jorgensen
Feng Tian

Abstract. Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $$\mathscr{H}_{E}$$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $$\mathscr{H}_{E}$$ we characterize the Friedrichs extension of the $$\mathscr{H}_{E}$$-graph Laplacian. We consider infinite connected network-graphs $$G=\left(V,E\right)$$, $$V$$ for vertices, and $$E$$ for edges. To every conductance function $$c$$ on the edges $$E$$ of $$G$$, there is an associated pair $$\left(\mathscr{H}_{E},\Delta\right)$$ where $$\mathscr{H}_{E}$$ in an energy Hilbert space, and $$\Delta\left(=\Delta_{c}\right)$$ is the $$c$$-Graph Laplacian; both depending on the choice of conductance function $$c$$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $$\mathscr{H}_{E}$$ consisting of dipoles. Now $$\Delta$$ is a well-defined semibounded Hermitian operator in both of the Hilbert $$l^{2}\left(V\right)$$ and $$\mathscr{H}_{E}$$. It is known to automatically be essentially selfadjoint as an $$l^{2}\left(V\right)$$-operator, but generally not as an $$\mathscr{H}_{E}$$ operator. Hence as an $$\mathscr{H}_{E}$$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $$l^{2}\left(V\right)$$.

Keywords: unbounded operators, deficiency-indices, Hilbert space, boundary values, weighted graph, reproducing kernel, Dirichlet form, graph Laplacian, resistance network, harmonic analysis, harmonic analysis, frame, Parseval frame, Friedrichs extension, reversible random walk, resistance distance, energy Hilbert space.

Mathematics Subject Classification: 47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 46N20, 22E70, 31A15, 58J65, 81S25.

Full text (pdf)